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Beliebt Algebra >

z^6-(3-2i)z^3+2-2i=0

Lösung

z^{6} - (3 - 2 i) z^{3} + 2 - 2 i = 0

Lösung

z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = \frac{1 + 3}{2} + i \frac{1 - 3}{2}, z = \frac{1 - 3}{2} + i \frac{1 + 3}{2}, z = - 1 - i

Schritte zur Lösung

z^{6} - (3 - 2 i) z^{3} + 2 - 2 i = 0

Schreibe

z^{6} - (3 - 2 i) z^{3} + 2 - 2 i

um:

(z^{6} - 3 z^{3} + 2) + i (- 2 + 2 z^{3})

(z^{6} - 3 z^{3} + 2) + i (- 2 + 2 z^{3}) = 0

Schreibe in der Standard Form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

z^{6} + (- 3 + 2 i) z^{3} + 2 - 2 i = 0

Schreibe die Gleichung um mit

u = z^{3}

und

u^{2} = z^{6}

u^{2} + (- 3 + 2 i) u + 2 - 2 i = 0

Löse

u^{2} + (- 3 + 2 i) u + 2 - 2 i = 0 : u = 1, u = 2 - 2 i

u = 1, u = 2 - 2 i

Setze

u = z^{3} w i e d er e in,

löse für

z

Löse

z^{3} = 1 : z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i

Löse

z^{3} = 2 - 2 i : z = \frac{1 + 3}{2} + i \frac{1 - 3}{2}, z = \frac{1 - 3}{2} + i \frac{1 + 3}{2}, z = - 1 - i

Die Lösungen sind

z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = \frac{1 + 3}{2} + i \frac{1 - 3}{2}, z = \frac{1 - 3}{2} + i \frac{1 + 3}{2}, z = - 1 - i

Beliebte Beispiele

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(x^2-x-3)^2-2(x^2-x)+3=0

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